Problem: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{-2x^2 - 8x + 64}{9x^3 + 117x^2 + 360x}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {-2(x^2 + 4x - 32)} {9x(x^2 + 13x + 40)} $ $ n = -\dfrac{2}{9x} \cdot \dfrac{x^2 + 4x - 32}{x^2 + 13x + 40} $ Next factor the numerator and denominator. $ n = - \dfrac{2}{9x} \cdot \dfrac{(x + 8)(x - 4)}{(x + 8)(x + 5)}$ Assuming $x \neq -8$ , we can cancel the $x + 8$ $ n = - \dfrac{2}{9x} \cdot \dfrac{x - 4}{x + 5}$ Therefore: $ n = \dfrac{ -2(x - 4)}{ 9x(x + 5)}$, $x \neq -8$